To enlighten my leaders with the theory of vacant places in Bridge and its effectiveness in declarer play when declarer is faced with the dilemma of either going for a finesse or drop in the making of his contract or finding of the opponent who holds the key card, I intend to cover the subject in 3 parts in the following 3 weeks assuring my readers that at the end they would be in a better position to make that critical decision.
In the vacant places theory or Bridge, it is the number of cards dealt to each defender that matters, not the number each holds at the critical moment. So rule 1 of theory of vacant places states that "When the distribution of one suit (or more) is completely known, the probability that an opponent holds a particular card in any other suit is proportional to the number of vacant places remaining in his hand eg in a 7 spade contract.
Trump distribution becomes known and count of vacant places puts west with 10, east with 13 (west holding all three trumps to east's Nil). The odds of east holding QC is 13:10. So in the effort to eliminate the red suits before committing in clubs to find CQ, suppose west discards a diamond on the third round of hearts. Now west had 3 spades originally and 2 hearts only = 5 known cards. He has 8 vacant places to east's 7 who has 6 hearts, no spades leaving 7 vacant places. Now the odds of west holding QC increase 8 to 7.
Vacant places method can also be used when not only a distribution of a side suit is known, but from the critical suit itself, provided the location of all 9 small cards in the suit are known eg with AK104 Opposite Q73 Both defenders follow to the K & Q with low cards and west plays a low card again on the third round. In this situation it is as though the hearts J and the remaining small heart belonged to different suits, since neither would play the jack as long as he held a lower heart to play.
So knowing the complete distribution of the small known cards of hearts in that suit apart from JQKA, we use vacant places calculation to locate JH. West with 3 small hearts has 10 vacant places while east's with 2 hearts has 11 vacant places. The ratio of JH being with east being 11:10, we go for the drop. The probability ratio has increased to 52.4% for the 3-3 break. So our next rule states "The critical suit may be included in a vacant place calculation when the location of all the small cards are known from 2 to 10".eg.
A1098 Opposite K24
Both opponents follow with small cards when KH is cashed. So only 2 small cards remain. If west has the Queen , he will have 10 vacant places to east's 11, when he follows with a low heart, making odds of 11:10 'in east's favour holding the JH, which gives probability of 3-3 break at 52.24%
Now supposing with Contract is 3NT. Lead is roman lead of 10S promising an honour above. Playing low in dummy, king pops up from east, taking it with ace you need to find DQ to run all 5 tricks to make your contract with 2 spades, 5D, 2 clubs for 9. Will diamonds, break 2-2 being combined 9? Or is it the 3-1 break for the QD? That the vital issue.
One thing is known - spade distribution. West holds 6, east 1 giving west 7 vacant places to east's 12 vacant places thereby giving 7:12 ratio for east holding the queen of diamonds. So when you play to AD, low cards follow and on return of a diamond, east follows with low diamond. Out of the 3 low cards, all 3 have appeared in this critical suit.
So placing east with 1 spade and 2 diamonds, east has 10 vacant places to west's, 6 spades plus 1 diamond = 6 vacant places. The odds are 10:6 or 5:3 that east holds Q of diamonds. Therefore, the finesse offers a 62.5% probability of success.
Another example in a 4H contract after east's 3D pre-empt.
West leads QD in the contract of 4H. You have a loser in each side suit and need to bring the trumps suit in without loss. East overtakes QD with KD to your AD west most probably had 1 to east's 7. Distribution of side suit is known. Chances of KH being with west with 12 VP to east's 6 VP's is 12:6. When you continue with QH, west plays 5- since all little cards are known in the critical suit west has 11 VP to east's 6. The odds are 11:6 that west hold KH. The probability of the K falling on 11 combined trumps is no longer in favour of the normal high percentage break of 1-1 (52%) but (48%) of being 2-0, in favour of the finesse. The percentage has increased to 65% for king being doubleton with west from the 48% standard percentage. Even if diamonds are 6-2, the odds favour the finesse. (To be continued)
North South North South North South
AK762 QJ983 Q73 A64 Q75 K9
KQ5 A8 K105 97 A10763 QJ9842
7 A4 A965 KJ1083 82 A65
K1032 AJ74 1062 AK5 1095 A3